grammar...is it fine add IS in between these words?

The courier office is continuously messing up on account of old address continuously messing up on account of old address
The courier office continuously messing up on account of old address continuously messing up on account of old address
Whether the verb is needed or not in the above sentences.

'is' is necessary because the verb here is 'is ... messing up'.
'Continuously' is an adverb which tells us HOW the courier office is messing up.
Actually, 'continually' would be the correct word to use. 'Continuous' means that something is happening all the time without a break. 'Continual' means that something continues to happen (but that each event may be separate).
Water may be said to flow continuously. Traffic may flow continuously past a point but the individual cars pass the point continually.
http://www.dailywritingtips.com/continuous-or-continual/
The correct statement would be: "The courier office is continually messing up on account of [because of] an old address/old addresses."
It is better English, also, to use 'because of' rather than 'on account of' because:
a) 'on account of' is jargon. The idiom arose from the use of the word in the context: "Please give me an account of your actions." (Meaning, 'please give me a detailed list of your actions so that I know the reasons'). "To call someone to account" means "to ask someone to explain the reasons for and detail of what they have done." Using the idiom "on account of" to mean "because of" is inaccurate, unnecessarily long and somewhat pompous (it sounds more 'educated' or 'official' than it really is).
b) the statement is in the context of a business and the word 'account' has a specific meaning here which is different from the meaning of the phrase 'on account of'.
'To mess [something] up' is a colloquial or slang expression. If you made this statement in informal speech, for example, in a business meeting, it would be fairly acceptable to use the verb 'to mess up'. If you are writing a report, you should avoid all such colloquial expressions.

sir, the actual information is the following
The address issue has been disturbing our center again and again. The courier office is continuously messing up on account of old address and as a result a lot of problems arise now. Sir, We are not responsible or liable for any issues in futureif this address problems continuous.
the above letter is written by me to my supervisor urging him to take actions.
please advise me

I would write it as follows:
[b]Dear Sir,
The problem of incorrect addresses has been the cause of constant delays in our center. The courier office continues to cause problems because it is using outdated addresses. We cannot be held responsible or liable for any issues in future if this address problem persists.[/b]
1) Be more specific in the first sentence: "the problem of incorrect addresses" instead of "the address issue".
2) Change the wording of the second sentence so that it is more direct and less 'jargon'. 'continues to cause problems' rather than 'is continuously messing up'; 'because of' instead of 'on account of'; 'as a result a lot of problems arise now' is redundant: you have already described the problems you are having in the first sentence.
3) 'We cannot be held responsible' instead of 'We are not responsible' (it is your supervisor who holds you responsible and you are disputing his view and telling him that he cannot do so in future). 'this ... problem' (and 'this issue') is a single thing (even though it involves a number of addresses), so 'this' and 'address problem' ('address', in this case, is an adjective describing the problem and 'this' refers to the '(address) problem'). 'persists' instead of 'continues' - I assume that 'continuous' was a typo - because I have already used 'continues' in the previous sentence and it sounds better to use a different word with the same meaning.

Hello Manasha. I'm no native English speaker; I'm a Filipino. But I have been working online for more than two years now, first as an editor and second as a writer. The previous responder said "has been" should be used to show the continuity of the action. I agree with him/her for that matter. This is what we call the PRESENT PERFECT PROGRESSIVE tense. You can do further study on this topic on your own. Good luck.

One should certainly use 'has been' to express an action which happened in the past and, because it happened 'continually', a past continuous tense, rather than a past perfect tense would normally be used. (Different grammarians have given different high-falutin' names to parts of speech and there is NO standardisation - I use the terms which I have learned and which seem to make the most sense to me!)
The issue, however, is not whether the action happened in the past or is happening now (and what tense should be used) but whether one should use the auxiliary 'to be' at all ... "'To be' or NOT 'to be', that is the question!"

I was born and bred in England. My father was a writer and editor for a major publisher and my mother was a teacher. Although I never became a teacher (of English, at least), the proper use of English has always been a particular interest of mine.

I am from India and my mother tongue is Hindi.I have learned English more from reading English newspapers and books and not in schools and colleges.In the same way I have studied English grammar.So I may be wrong here,but after going through the two sentences,I think,the correct sentence will be,'The courier office HAS BEEN continuously messing up on account old address.'
'Messing up' is a verbal phrase.Here the act of messing up had started earlier- in the the past- and it is going on in the present as well.That means,the act has not ended.It has been continuously going on.So here 'has been' should be used.
Although I may be wrong.I am waiting for some native English speaker's response.

We don't know (from the context) whether the action was continual in the past (but has now ceased) or whether it still continues at this moment. It is safe to assume (without evidence to the contrary) that the courier office is STILL sending parcels to the wrong addresses and therefore "is continually messing up" is what is intended.
Please see my response below.

If you strictly follow grammar it would be 'has been continuously' in stead of 'is continuously' as the process is going on for at least a couple of days and hence should fall in the category of 'Present Perfect Continuous' tense. But 'is continuously' is also correct and no one would be able to pen through it.